Random process; white noise PSDF and autocorrelation

Random process

A random (stochastic) process $X(t,s)$ is a collection (ensemble) of time functions, one for each outcome $s$ of a random experiment. At a fixed time $t_1$, $X(t_1)$ is a random variable; for a fixed outcome it is an ordinary waveform (a sample function). Example: thermal noise voltages across many identical resistors; the received signal in a communication channel. It is described by its mean $m_X(t)=E[X(t)]$, autocorrelation $R_X(t_1,t_2)$ and PSD.

White noise

White noise is an idealised random noise whose power spectral density is flat (constant) over all frequencies, like white light containing all colours. Thermal and shot noise are nearly white up to very high frequencies ($\sim 10^{12}$ Hz), so the model is accurate for any practical receiver bandwidth. It is usually also assumed Gaussian, zero-mean (AWGN).

PSD (two-sided):

$$S_W(f)=\frac{N_0}{2}\quad\text{for all } f$$

where $N_0=kT_e$ (W/Hz), $k$ = Boltzmann constant and $T_e$ = equivalent noise temperature.

Autocorrelation (inverse Fourier transform of PSD, using $\mathcal{F}^{-1}\{1\}=\delta(\tau)$):

$$R_W(\tau)=\int_{-\infty}^{\infty}\frac{N_0}{2}e^{j2\pi f\tau}df=\frac{N_0}{2}\,\delta(\tau)$$

   S_W(f)                      R_W(tau)
     |                             ^  (N0/2) delta
 N0/2+-----------------            |
     |                             |
 ----+-----------------> f   ------+------> tau
     0                             0

Meaning: $R_W(\tau)=0$ for every $\tau\ne0$, so any two samples of white noise, however close, are uncorrelated (and independent if Gaussian). The total power $R_W(0)=\int S_W(f)df$ is infinite, which is why white noise is only a model; after any filter with finite bandwidth the power becomes finite.